Theorem bj-cbv1v 32729

Description: Version of cbv1 2267 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv1v.1	|- F/ x ph
bj-cbv1v.2	|- F/ y ph
bj-cbv1v.3	|- ( ph -> F/ y ps )
bj-cbv1v.4	|- ( ph -> F/ x ch )
bj-cbv1v.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
bj-cbv1v	|- ( ph -> ( A. x ps -> A. y ch ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y)

Proof of Theorem bj-cbv1v

Step	Hyp	Ref	Expression
1		bj-cbv1v.2	. . . . 5 |- F/ y ph
2		bj-cbv1v.3	. . . . 5 |- ( ph -> F/ y ps )
3	1, 2	nfim1 2067	. . . 4 |- F/ y ( ph -> ps )
4		bj-cbv1v.1	. . . . 5 |- F/ x ph
5		bj-cbv1v.4	. . . . 5 |- ( ph -> F/ x ch )
6	4, 5	nfim1 2067	. . . 4 |- F/ x ( ph -> ch )
7		bj-cbv1v.5	. . . . . 6 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
8	7	com12 32	. . . . 5 |- ( x = y -> ( ph -> ( ps -> ch ) ) )
9	8	a2d 29	. . . 4 |- ( x = y -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
10	3, 6, 9	cbv3v 2172	. . 3 |- ( A. x ( ph -> ps ) -> A. y ( ph -> ch ) )
11	4	19.21 2075	. . 3 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
12	1	19.21 2075	. . 3 |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
13	10, 11, 12	3imtr3i 280	. 2 |- ( ( ph -> A. x ps ) -> ( ph -> A. y ch ) )
14	13	pm2.86i 109	1 |- ( ph -> ( A. x ps -> A. y ch ) )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-cbv1hv  32730